The tangent splash in PG(6,q)

Let B be a subplane of PG(2,q3) of order q that is tangent to ℓ∞. Then the tangent splash of B is defined to be the set of q2+1 points of ℓ∞ that lie on a line of B. Tangent splashes of PG(2,q3) are all projectively equivalent, and are equivalent to GF(q)-linear sets of rank 3 and size q2+1. In the Bruck-Bose representation of PG(2,q3) in PG(6,q), we investigate the interaction between the ruled surface corresponding to B and the planes corresponding to the tangent splash of B. We then give a geometric construction of the unique order-q-subplane determined by a given tangent splash and a fixed order-q-subline.